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Random Walk on the Range of Random Walk

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Abstract

We study the random walk X on the range of a simple random walk on ℤd in dimensions d≥4. When d≥5 we establish quenched and annealed scaling limits for the process X, which show that the intersections of the original simple random walk path are essentially unimportant. For d=4 our results are less precise, but we are able to show that any scaling limit for X will require logarithmic corrections to the polynomial scaling factors seen in higher dimensions. Furthermore, we demonstrate that when d=4 similar logarithmic corrections are necessary in describing the asymptotic behavior of the return probability of X to the origin.

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  1. Department of Statistics, University of Warwick, Coventry, CV4 7AL, UK

    David A. Croydon

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  1. David A. Croydon
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Croydon, D.A. Random Walk on the Range of Random Walk. J Stat Phys 136, 349–372 (2009). https://doi.org/10.1007/s10955-009-9785-2

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  • DOI: https://doi.org/10.1007/s10955-009-9785-2

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